Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 301 / 2 / b / ii / Solution

The scalar kinetic operator is proportional in momentum space to $k^2(\tilde{c}{k}^2-m^2)$. Its inverse has the partial-fraction decompositionup to the overall convention-dependent factor allowed in the question. Thus $\tilde{c}=\tilde{a}+\tilde{b}$. Under the physical relation $\tilde{b}=-\tilde{a}$, one has $\tilde{c}=0$, so the extra massive pole and its ghost residue disappear.